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TIMO 2017 - Senior Secondary (Grades 10-12) - Preliminary Round - Paper 5

Question 1

A box contains 2015 balls coloured red, green, blue, yellow or black. Any draw of 1800 balls guarantees at least one ball of every colour. What is the minimum number of balls to draw to guarantee three different colours?
  1. 1797
  2. 1798
  3. 1368
  4. 1367

Question 2

Find the next term in $1024,0,512,256,384,320,352,\ldots$.
  1. 336
  2. 672
  3. 448
  4. 310

Question 3

Primes $x,y$ satisfy $x^2-y^3=1$. Find $x+y$.
  1. 3
  2. 4
  3. 5
  4. 6

Question 4

Six teams $A,B,C,D,E,F$ play a five-day tournament. Each team plays once per day and plays each other team exactly once. On day 1, $A$ beats $B$; on day 2, $D$ beats $C$; on day 3, $E$ beats $A$; on day 4, $B$ beats $C$. Which team plays $E$ on day 5?
  1. Team F
  2. Team B
  3. Team C
  4. Team D

Question 5

Seven people make these statements about today: A: Yesterday was Wednesday. B: Tomorrow is not Tuesday. C: Tomorrow is Wednesday. D: Today or yesterday is Tuesday. E: Today is Tuesday. F: Today is neither Monday, Tuesday nor Sunday. G: Today is neither Tuesday nor Saturday. Exactly three statements are true. What day is today?
  1. Monday
  2. Wednesday
  3. Friday
  4. Sunday

Question 6

Express $\left(\frac12-\frac{\sqrt3}{2}i\right)^{1111}$ in the form $a+bi$.
  1. $\frac12+\frac{\sqrt3}{2}i$
  2. $\frac12-\frac{\sqrt3}{2}i$
  3. $-\frac12+\frac{\sqrt3}{2}i$
  4. $-\frac12-\frac{\sqrt3}{2}i$

Question 7

$x^4-x^3+ax^2+bx+3=0$, with constants $a,b$, has a double root at 1. How many roots have nonzero imaginary part?
  1. 0
  2. 1
  3. 2
  4. 3

Question 8

Compute $\displaystyle\sum_{k=2}^{10}\frac1{k^2-1}$.
  1. $36/55$
  2. $72/55$
  3. $5/11$
  4. $10/11$

Question 9

Find the remainder on dividing $x^{2015}-20x-15$ by $x^2-1$.
  1. $19x+15$
  2. $19x-15$
  3. $-19x+15$
  4. $-19x-15$

Question 10

Factorise $x^2-3xy+2y^2+x-4y-6$.
  1. $(x-y-3)(x-2y+2)$
  2. $(x+y-3)(x+2y+2)$
  3. $(x-y+3)(x-2y-2)$
  4. $(x+y-3)(x-2y+2)$

Question 11

Positive real numbers $a,b,c$ satisfy $a^2+b^2+c^2\le24$. Find the maximum of $a^{2/3}+b^{2/3}+c^{2/3}$.
  1. 4
  2. 5
  3. 6
  4. 7

Question 12

For $-3\le x\le3$, find the maximum of $x+\sqrt{9-x^2}$.
  1. 3
  2. $3\sqrt2$
  3. 6
  4. $3\sqrt3$

Question 13

Given $\log 2=0.301$, evaluate $\log\left(\sum_{n=0}^{2015}\binom{2015}{n}\right)$ to three decimal places.
  1. 606.505
  2. 606.515
  3. 606.525
  4. 606.535

Question 14

Find the smallest positive integer $x$ satisfying $8x\equiv9\pmod{17}$ and $3x\equiv5\pmod7$.
  1. 64
  2. 65
  3. 66
  4. 67

Question 15

Both the real number $a$ and $\sqrt{23-8\sqrt7}+a\sqrt{23+8\sqrt7}$ are rational. Find $a$.
  1. 0
  2. 1
  3. 2
  4. 3

Question 16

Two tangents to $x^2+y^2=1$ pass through $P(3,4)$. Find the sum of their slopes.
  1. 2
  2. -3
  3. -2
  4. 3

Question 17

The altitudes $AD,BE$ of acute triangle $ABC$ intersect at $H$. Join $D$ and $E$. How many pairs of similar triangles are there?
  1. 10
  2. 2
  3. 8
  4. 4

Question 18

A triangle has side lengths $a,b,c$ satisfying $0\le a\le6\le b\le8\le c\le12$. Find its perimeter when its area is maximal.
  1. 24
  2. 23
  3. 22
  4. 25

Question 19

Find the shortest distance from $(5,-10)$ to the line $4y=3x$.
  1. 10
  2. 11
  3. 8
  4. 9

Question 20

$TA$ and $DC$ are tangents to the circle at $A,C$. Points $T,B,C$ are collinear. Given $\angle ATB=50^\circ$ and $\angle BCD=75^\circ$, find $\angle ABC$.

Question illustration
  1. $75^\circ$
  2. $77.5^\circ$
  3. $80^\circ$
  4. $72.5^\circ$